04 May 2009

Bears, pigs, and the like

The blog's been slow. I've been off writing real mathematics, thinking for and preparing for the class I'm teaching this summer, and so on. But I'm still here!

And while I'm here, you should read Chad Orzel on the faulty thermodynamics of children's stories. In the story of Goldilocks and the three bears, one would expect that the papa bear is the largest, then the mama bear, and then the baby bear. Furthermore, you'd think that the larger the bear, the larger the bowl of porridge, and the slower it should cool off. But it doesn't seem to work that way! Read the comments come up with some interesting explanations.

Exercise for the scientifically-inclined reader: comment on the physical implications of the Three Little Pigs.

Exercise for the not-so-scientifically-inclined reader: what's with all the animals coming in threes?

It's not just animals. Want to jump off a cliff simultaneously with someone else? You "go on three". Some languages only have words for specific numbers up to three. It seems like we count up to three subconsciously. Suppose you and someone else walk past a table where some people are eating lunch. An hour later, ask your companion how many people were at that table. I'll bet that you see a dramatic drop off in accuracy when the number of people at the table reaches four. Our minds seem to have some sort of fuzzy cutoff point at three dividing the built-in numbers from the ones we've constructed. Jokes come in threes (an engineer, a scientist, a mathematician) etc.

I suspect that one of the reasons joke things come in threes is that two is the minimum number of things you need to establish a pattern. Hence, three is the minimum number of things you need to establish a pattern and then break it.

What I want to know is, why do axioms often come in threes? The open-set axioms for a topology, the axioms for a sigma-algebra, the axioms for a partial or total order...